#7092555

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,28,23,27,37,40)(19,29,32,26,31,44,21,24)(20,25,36,42,35,43,47,48)(30,45,34,41,33,46,38,39);(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,27,26)(20,33,35,30)(21,37,32,22)(25,41,43,39)(28,44,40,29)(34,47,38,36)(42,45,48,46);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,38)(22,36)(23,35)(24,39)(26,41)(27,43)(28,42)(29,45)(31,33)(32,34)(37,47)(40,48)(44,46);(11,16)(13,15)(17,21,22,19,23,32,37,31)(18,26,28,44,27,24,40,29)(20,34,36,33,35,38,47,30)(25,39,42,45,43,41,48,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(19,31)(20,35)(21,32)(28,40)(36,47)(39,41)(45,46)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ghi,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6999751);